Convergent meshfree approximation schemes of arbitrary order and ...

100 A. Bompadre et al. / Comput. Methods Appl. Mech. Engrg. 221–222 (2012) 83–103ln(L 2 −error)−8−10−12−14−16HOLMESB−SplineMLSFEM3 : 1ln(H 1 −error)−4−6−8−10HOLMESB−SplineMLSFEM2 : 1−18−12−20−5 −4.5 −4 −3.5 −3 −2.5 −2ln(h)−14−5 −4.5 −4 −3.5 −3 −2.5 −2ln(h)Fig. 16. L 2 - and H 1 -error convergence of the displacement field of an elastic membrane under sine load approximated with different types of basis functions: higher ordermax-ent functions (HOLMES), B-splines, moving least squares functions (MLS) and Lagrange polynomials (FEM).H 1 -error norms of the displacement field. The L 2 - and H 1 -errornorms are calculated with respect to the exact analytical solutionsfor plate bending problems found in [30]. We also compute the H 1 -error norm restricted to the boundary of the plate domain wherethe boundary conditions are imposed. The convergence of theH 1 -error norm restricted to the constrained boundaries assuresthat boundary conditions are imposed correctly using Lagrangemultipliers. The L 2 - and H 1 -error norms are computed approximatelyusing 2:5 10 5 uniformly distributed sampling points inthe plate domain.The order of convergence of the error norms computed in theplate examples is reported in Table 6 for two different values ofc. It is noteworthy that the order of convergence is consistentlyclose to, or higher than, the theoretical convergence order of 2for both values of c and throughout the numerical examples, whichdiffer strongly with respect to loading and boundary conditions. Asin the case of second-order problems, a comparison of Table 6 andFig. 13 reveals that the general trends in one dimension carry overto higher dimensions. For reasons of brevity, it is not possible to reportthe full convergence plots and deformed configurations for allTable 6Convergence rates for different displacement error norms computed in several numerical examples of a Kirchhoff plate subjected to different loads with different boundaryconditions (the abbreviation SS refers to simply supported boundary conditions). Two different values of c are used to compute the HOLMES approximation functions.L 2 H 1 H 1 boundaryc ¼ 0:3 c ¼ 0:8 c ¼ 0:3 c ¼ 0:8 c ¼ 0:3 c ¼ 0:8Clamped – concentrated load 2.17 2.41 2.03 2.16 3.44 2.35Clamped – uniform load 2.18 2.45 2.12 2.33 4.10 2.17SS on 2 edges – uniform load 2.26 2.39 2.23 2.36 2.38 2.34SS on 4 edges – concentrated load 2.15 2.32 1.99 1.85 2.27 1.94SS on 4 edges – uniform load 2.13 2.65 2.04 2.31 2.17 2.60SS on 4 edges – sine load 2.14 2.79 2.11 2.49 2.22 2.9710 −2 h10 −3L 2H 1H 1 boundary η = 2.1610 −4η = 2.41Error norms10 −510 −6η = 2.3510 −710 −810 −910 −2 10 −1Fig. 17. Elastic Kirchhoff plate clamped on all four edges and subjected to a unit centered concentrated load. Convergence of the L 2 - and H 1 -error norms of the displacementfield (left) and deformed configuration for the finest nodal distribution (55 55 nodes) used in the convergence analysis (right). The order of convergence g for each errornorm is also reported.